Circling the Square

This simple, fun and elegant method to circling the square (an attempt to make a circle equal in area to that of a given square or visa versa) using a straightedge and compass only, has an accuracy that exceeds 99.9%.

1. Place eight interlocking circles (L/2) along a straight line

2. To create the square (L²):
First, use section (L/2) to draw the center circle, followed by four outside circles and then the square.

3. Draw a line through points A and B and circle R1 as shown

4. Circle R2

5. Circle R3 which is the radius

><>*<><

In this circling the square method the ratio of Earth to Moon diameter is 3.66... and matches the real thing!

Thank You toMichael Joycefor putting this into proper mathematical format for me.

"Here follows a mathematical proof of Gertjan’s discovery.

If the radius of each of the small 8 circles is 1, then the square will have sides of length 18 (2 x 9).
D4 represents the diameter of a circle, R1, R2, R3 the radii of other circles.

1- To calculate angle “a”
The diagram in the lower right quadrant shows an enlargement of the second small circle
cosine a = 0.5 divided by 1
= 1/2
Therefore a = 60 degrees.

Since the Earth and Moon are only approximately spherical; no single value can serve as a definitive radius.
For example, the Earth’s meaningful values range from 6355 to 6378 kilometres (3949 to 3963 mi).
However, the Earth and Moon will be exact spheres in an absolute reality.
Perhaps with ratio of diameters = 3.66943... : 1